Applications of Differentiation - Derivative as the Gradient of a Curve

A curve has the equation \(y = \sqrt{ax^2 - 3x + c}\), where \(a\) and \(c\) are constants.

1. Given that the curve passes through \((0, \sqrt{6})\) and that the tangent to the curve at \(x = \frac{3}{2}\) is parallel to the \(x\)-axis, find the values of \(a\) and \(c\).

Hence find

2. the coordinates of the points on the curve where the gradient of the curve is \(\frac{1}{2}\), and
3. the range of values of \(x\) where the gradient of the curve is positive.